9 Long-term Water Balance Modeling
In this Chapter, we investigate long-term mean catchment water balances from a large, regional perspective.
9.1 Introduction
The general water balance of a catchment can be written as
\[ \Delta S = P - E - Q \tag{9.1}\]
where \(\Delta S\) is net storage change in millimeters [mm], \(P\) is precipitation in mm, \(E\) is evaporation in mm, and \(Q\) is specific discharge in mm. Evaporation is when a substance is converted from its liquid into its vapor phase, independently of where it lies in nature (Miralles et al. 2020). This definition of evaporation encompasses evaporation from inside leaves (transpiration), evaporation from bare soils, evaporation from intercepted precipitation (interception loss), evaporation from open water surfaces, and finally, evaporation over ice- and snow-covered surfaces (often referred to as sublimation).
Over the period of a hydrological year and longer time scales, we expect \(\Delta S\) to be 0 since neither water storage nor destorage happens over longer periods. This would not be true for catchments where, for example, man-made storage infrastructure was built over the period under consideration or for catchments with ongoing glacier melt over a prolonged time. If \(\Delta S = 0\), the above Equation @ref(eq:WB1) can be rewritten as
\[ Q = P - E \tag{9.2}\]
Dividing by \(P\), we get
\[ \frac{Q}{P} = 1 - \frac{E}{P} \tag{9.3}\]
where \(Q/P\) can be called the runoff index and \(E/P\) is the evaporation index or evaporative fraction.
For a catchment, the annual mean \(E\) and \(Q\) are governed by the total water supply \(P\) and the total available energy, which is expressed as potential evaporation \(E_{pot}\) and which denotes the (atmospheric) water demand. If \(E_{pot}\) is small, the discharge \(Q\) usually is bigger than evaporation \(E\). Similarly, if the available radiative energy is very high, the water demand \(E_{pot}\) is very large and \(Q \ll E\) (Arora 2002). \(E_{pot}\) and \(P\) are thus the critical determinants of annual or longer timescale runoff and evaporation rates. Michael Budyko has termed the ratio \(E_{pot} / P\) aridity index (Budyko 1974).
As explained above, water demand is determined by energy. Solar radiation is the primary energy source for the earth-atmosphere system and the key driver of the hydrological cycle. At the earth’s surface, the net radiative flux \(R_N\) is the energy that is available for a) heating and cooling of the soil (ground heat flux), b) changing the phase of water (latent heat flux), and c) heating or cooling air in the boundary layer thus causing atmospheric dynamics (sensible heat flux).
This can be formalized with the following relationship
\[ R_{N} = H_{S} + H_{L} + \Delta H_{G} \tag{9.4}\]
where \(R_{N}\) is the net radiation [in W/m2 = kg/s3], \(H_{S}\) is the upward sensible heat flux, \(H_{L}\) is the latent heat flux and \(\Delta H_{G}\) the net ground heat flux. The latent heat flux is directly proportional to evaporation \(E\). Thus, \(H_{L} = L \cdot E\) where \(L = 2.5 \cdot 10^{6}\) J/kg [= m2/s2] is the latent heat of vaporization and \(E\) is the actual evaporation in [m/s]. As in the case of the water balance, at the annual or longer time scales, we can neglect the heat storage effect in the ground and get
\[ R_{N} = H_{S} + L \cdot E \tag{9.5}\]
The Bowen ratio is defined as the fraction of the sensible heat flux divided by the latent heat flux, i.e.
\[ \gamma = \frac{H_{S}}{H_{L}} = \frac{H_{S}}{L \cdot E } \tag{9.6}\]
By rearranging the terms, the long-term energy balance in Equation Equation 9.5 can simply be rewritten as
\[ R_{N} = (1 + \gamma)L E \tag{9.7}\]
Using the fact that \(R_{N} = L E_{pot}\), where \(E_{pot}\) is the potential evaporation, and dividing by precipitation, we can rewrite the above Equation 9.7 as
\[ \frac{E_{pot}}{P} = (1 + \gamma) \frac{E}{P} \tag{9.8}\]
where the left-hand side is called the aridity index, i.e. \(\phi = E_{pot}/P\) and \(E/P\) is called the evaporative fraction or evaporation index, as mentioned above. With this, Equation 9.8 can be written as a function of the Bowen ratio and the aridity index, i.e.
\[ \frac{E}{P} = 1 - \frac{Q}{P} = \frac{\phi}{(1 + \gamma)} \tag{9.9}\]
\(Q/P\) is again the runoff index. Since the Bowen ratio is also water supply and energy demand limited, it too is a function of the aridity index and we can thus rewrite Equation 9.9 as
\[ \frac{E}{P} = \frac{\phi}{1 + f(\phi)} = F[\phi] \tag{9.10}\]
The Budyko relationship thus allows for a simple parameterization of how the aridity index \(\phi\) controls the long-term mean partitioning of precipitation into stream-flow and evaporation and it is capable of capturing the behavior of thousands of catchments around the world. This explains its growing popularity over recent years (Berghuijs, Gnann, and Woods 2020).
Figure @ref(fig:budykoSpace) shows a plot of data from catchments in the US for which consistent long-term hydro-climatological data records are available. Individual catchments’ aridity indices are plotted against evaporative fractions, averaged over many years. The catchment data plots along the Budyko curve in the two-dimensional Budyko space as indicated in the Figure where the Budyko curve is defined as
\[ \frac{E}{P} = \left[ \frac{E_{pot}}{P} \text{tanh} \left( \frac{P}{E_{pot}} \right) \left( 1 - \text{exp} \left( - \frac{E_{pot}}{P} \right) \right) \right]^{1/2} \tag{9.11}\]
This non-parametric relationship between the aridity index and the evaporative fraction was developed by M. Budyko (Budyko 1951).
The Budyko space is delineated by the demand and supply limits. Catchments within the space should theoretically fall below the supply limit (\(E/P = 1\)) and the demand limit (\(E/E_{pot} = 1\)), but tend to approach these limits under very arid or very wet conditions (Berghuijs, Gnann, and Woods 2020). The data from the US shows that a large percentage of in-between catchment variability can be explained by the Budyko curve. After the seminal work Budyko in the last century, the evidence for a strong universal relationship between aridity and evaporative fraction via the Budyko curve has since grown. As catchment hydrology still lacks a comprehensive theory that could explain this simple behavior across diverse catchments Gentine et al. (2012), the ongoing debate about the the underlying reasons for this relationship continues (see e.g. (Padron et al. 2017; Berghuijs, Gnann, and Woods 2020)).
While almost all catchments plot within a small envelope of the original Budyko curve, systematic deviations are nevertheless observed from the original Budyko curve. Several new expressions for \(F[\phi]\) were therefore developed to describe the long-term catchment water balance with one parameter (see e.g. Budyko (1974); Sposito (2017); Choudhury (1999)). One popular equation using only 1 parameter is the Choudhury equation which relates the aridity index \(\phi\) to the evaporative fraction \(E/P\) in the following way
\[ \frac{E}{P} = \left[ 1 + \left( \frac{E_{pot}}{P} \right) ^{-n} \right]^{1/n} \tag{9.12}\]
where \(n\) is a catchment-specific parameter which accounts for factors such as vegetation type and coverage, soil type and topography, etc. (see e.g. Zhang et al. (2015) for more information). In other words, \(n\) integrates the net effects of all controls of of the evaporative fraction other than aridity. The Figure @ref(fig:ChoudhuryEquationStateSpace) shows the control of \(n\) over the shape of the Budyko Curve.
9.2 Data and Methods
9.2.1 Data
A large number of geospatial data were collected for the Central Asia region. The domain of interest was defined as 55 deg. E - 85. deg. E and 30 deg. N - 50 deg. N.. Shapefiles from the large river basin were retrieved from the Global Runoff Data Center and extracted for the following basins: Amu Darya, Chu, Issy Kul, Murghab-Harirud, Syr Darya and Talas. Where necessary, the polygons of the downstream flat areas were corrected to account for man-made water transfers via large canal systems and corresponding flow alterations across basins there. These large river basins define the area of interest (AOI).
For the selected basins, the WMOBB River Network data was extracted from the layers wmobb_rivnets_Q09_10 (containing line sections representing an upland area above 4’504 km2), wmobb_rivnets_Q08_09 (containing line sections representing an upland area between 1’150 km2 and 4’504 km2) and wmobb_rivnets_Q07_08 (containing line sections representing an upland area above between 487 and 1’150 km2) (GRDC, Koblenz, Germany: Federal Institute of Hydrology (BfG). 2020). Permanent water bodies and courses were taken from the global HydroLakes Database (Messager et al. 2016). Information on land cover were taken from the Copernicus Global Land Service: Land Cover 100m: collection 3: epoch 2019: Globe data (Buchhorn et al. 2019). The NASA SRTM digital elevation model 1 Arc-second (30 m) global product was used as a DEM (“NASA Shuttle Radar Topography Mission (SRTM)” 2013).
In total, data from 277 gauging stations from Afghanistan, Kyrgyzstan, Kazakhstan, Uzbekistan and Tajikistan could be obtained from the local Hydrometeorological Organization, public reports and the Soviet compendia Surface Water Resources, Vol 14 Issues 1 and 3. Except for the Afghan stations, all stations were manually located in a Geographic Information System (GIS) using the relevant Soviet Military Topographic maps (1:200’000) from the corresponding region. The maps were downloaded from https://maps.vlasenko.net and subsequently geo-referenced in QGIS (QGIS Development Team 2021). Data from northern Afghan rivers’ stream flow characteristics and the location of these gauging stations was taken from (Olson and Williams-Sether 2010).
For each gauge, the contributing area was delineated in R with the WhiteboxTools v2.0.0 and long-term norm mean discharge was obtained over variable observation periods between 1900 and 2018 was acquired. For a few selected stations, monthly and decadal time series data are available over the entire observational record. The FLO1K, global maps of mean, maximum and minimum annual stream flow at 1 km resolution from 1960 through 2015 were retrieved (Barbarossa et al. 2018). The goodness of the FLO1K product in the Central Asia domain was validated at the locations of the 277 gauges through linear regression.
Geospatial information on glaciers was taken from the Randolph Glacier Inventory (RGI) 6.0. Information from 16’617 glaciers was retrieved, together with glacier length, thickness and glacier thinning rates Hugonnet et al. (2021).
The CHELSA V21 global daily high-resolution climatology, available from 01-01-1979 until 31-12-2011 was processed over the Central Asia domain to map climate trends, including on temperature, precipitation, snow fraction. The data is available upon request from this site: https://chelsa-climate.org Karger et al. (2021). The CHELSA V21 product is corrected for snow undercatch in the high elevation ranges and thus is able to better represent actual high mountain precipitation than other available global climatologies (Beck et al. 2020). The aridity index (AI) fields were taken from the bio-climate CHELSA V21 data set and compared with the CGIAR AI product (Trabucco and Zomer 2019). Data on an additional 70 bio-climatic indicators were downloaded from the CHELSA V21 1980 - 2010 climatology and statistics extracted for each of the 277 gauged catchments, together with the AI.
High-resolution crop disaggregated irrigated areas were mapped over the entire Central Asia domain (Ragettli, Herberz, and Siegfried 2018). Like this 30 m crop maps were produced with Google Earth Engine using unsupervised classification for the years 2016 - 2020. Vector information on the irrigation systems in the Chu and Talas River basins as well as from the Uzbek Fergana Oblast, including the land cadaster there, are available.
Finally, data from the GOODD data set was used to retrieve information from 88 dams in the region of interest (Mulligan, Soesbergen, and Sáenz 2020).
9.2.2 Methods
A strategy for hydrological modeling of the regional Central Asian hydrology using the Budyko framework was devised. The Budyko principle posits that, over the long-run, runoff at a particular location is governed by the long-term availability of water (supply) and energy (demand) there (Budyko M., 1974). Under this assumption, the evaporative fraction of a basin, i.e. the long-term mean actual evaporation divided by long-term mean precipitation, can be expressed as a function of the aridity index (long-term mean potential evaporation divided by long-term mean precipitation).
9.3 Results
This Section is currently under active development. Please check back later.
9.4 Discussion and Conclusions
This Section is currently under active development. Please check back later.